Quadratic variation of c\`adl\`ag semimartingales as a.s. limit of the normalized truncated variations

Abstract

For a real c\`adl\`ag path $x$ we define sequence of semi-explicit quantities, which do not depend on any partitions and such that whenever $x$ is a path of a c\`adl\`ag semimartingale then these quantities tend a.s. to the continuous part of the quadratic variation of the semimartingale. Next, we derive several consequences of this result and propose a new approach to define F\"ollmer's pathwise integral.